Research: sharp COPY64 and COPY71 for all-nine-factor uniform P112 columns

[bitplane]

Rescaling all 52 free entries of one P112 column creates a rank-nine masked completion. This proves that every exact nine-term complex factorization still has separate COPY cost at least 64, with sharp power floor sum rho_j^q >= 8*8^q for every 0<q<=1. When all nine terms have nonzero original coordinate projections on both sides, the sharper bounds 71 and 8*8^q+7^q are also attained.

The family is K_lambda = XX^T + lambda*(X X_v^T - 4e_v)e_v^T, for every one of the 112 live columns and every nonzero complex lambda. Lambda=-1 clears the free column entries. All nine factors may vary, with ordinary transpose; the transposed row family follows. Only research/p112-uniform-column-rank9/ is added.

Mathematical contribution and credit

Arham Tahir's #1591, pinned at 264b8dcfc56584797ebf36d69ee35de18313fd27, handles one changed odd entry. Its complete punctured support classification is explicitly credited and independently replayed here; no priority for that lemma is claimed. The new 52-entry family has the same anchor on both sides, so singleton terms can pair at zero cost and the predecessor's distinct-anchor charge does not apply. The written proof resolves both zero-cost residuals using the actual coefficient matrix. Cross-duality permits at most one cost-seven mixed term, giving the sharp all-nine-mixed floor 71.

Gareth Davidson's #1468 and Hanson Sun's #1613 cover arbitrary column amplitudes while mixing two or three coordinate modes, respectively. The latter head is a34723742801f877c9636ed72f176a7ffa7bf005 and credits Andrey Mas's parallel claim. This package restricts amplitudes but frees all nine factors. #1612's separate COPY71 result fixes nine simplex normals and all-pairs exactness, a different domain. The multiline and compensation lineage, Mihir Srivastava's #845 geometry, DreamingOfClouds' #717/#446/#595 model, GamingPuzzled/#620, themilkman216/#968 and woklerx/#1077/#1180 are retained in NOTICE.md and the unchanged source copies.

Validation

python3.12 -I -B scripts/research_package.py --package research/p112-uniform-column-rank9

The complete registered replay passed:

• All 57,232 puncture/character cases, the unpunctured classification, and 13,552 support-class pairings.
• Fourteen exact Gaussian-rational factorizations, checking 175,616 matrix entries, including dense nonreal changes and attaining COPY64/COPY71 examples.
• A separate geometry/rank implementation and 37,632 independently computed integer compensation entries. Dropping the final paid term suggests COPY63 but corrupts a constrained diagonal.
• Eight external corruptions rejected for the intended reasons, plus optimized read-only relocation. Both checkers also passed isolated read-only Python 3.14 execution with identical primary receipt c9c67a0d484f8e9cbc05d9aab592b57df5eb3a70bf9ccc53d0ebbc7d2ccacb79.

Source pins, authenticated remote Git blobs, metadata, clean patch application and changed-package discovery passed. Mathematical commit: 6b05c428fcfdc5d860c30baa00d3b6fdc41da050.

Assignment and limits

One OS-random selection, nonce c3d61d9333f4d9682ef1c24c486cd260, chose revision-2 complex-mechanisms / complex-masked-decoder; both prescribed starting checks passed. Base ef5f39403610d231a0c27e8a6b4f08c10209bcd6. Priority revision 3 was read on resumption; the assignment was retained. Final upstream 1f7a934975e5eacaf83511c9ac870263bf095e05 preserves the mathematical inputs, selected record and replay hook. Final overlap covered 594 open PRs and 293 discussion records with complete pagination; the closest predecessor heads were unchanged. Scoped claim and finding.

This is a scoped COPY-component obstruction with a written quantified proof and exact finite premises. It does not supply a physical decoder, whole local paid profile, 5% ceiling-gate improvement or new kappa. Arbitrary amplitudes, multiple columns, ten or more terms, changed normalization and shared COPY billing remain outside scope. No selected cold replay, hosted CI success, independent human review or proof-assistant verification is claimed. Structured next questions cover those larger factor families and the missing full frame-path invoice.

Prepared by Gareth Davidson (@bitplane), with substantial OpenAI Codex assistance. Apache-2.0 with retained predecessor attribution.

[Mister-Meeseeks]

Batch review at 9f9295b1eb82056f31f4f1ac2b30ede0d7f8ca07.

Merged to main, preserving the original contributor commits and package attribution.

Reviewed scope: Uniform single-column rank-nine P112 completions: COPY64 and power-profile floor for every nonzero complex correction, COPY71 when all nine factors project nontrivially. Complete support-kernel classification and compensated branches; physical costs and broader masked completions remain outside scope.

The pinned scoped command passed; the batch ledger contains its command, input hashes, environment and log. This incorporates the research/tooling contribution and does not change the selected κ or discharge the inherited multiplication-theorem assumptions.

Batch review, attribution and evidence. Thank you for the contribution.